The HOMFLY-PT Polynomial
The HOMFLY-PT polynomial is a knot and link invariant. Confusingly, there are several variants depending on exactly which relationships are used to define it. All are related by simple substitutions.
To compute the HOMFLY-PT polynomial, one starts from an oriented link diagram? and uses the following rules:
is an isotopy invariant (thus, unchanged by Reidemeister moves).
Let , , and be links which are the same except for one part where they differ according to the diagrams below. Then, depending on the choice of variables:
- . (Sometimes is used instead of )
- Using three variables: .
From the rules, one can read off the relationships between the different formulations:
- , .
The HOMFLY polynomial generalises both the Jones polynomial and the Alexander polynomial (equivalently, the Conway polynomial?).
To get the Jones polynomial, make one of the following substitutions:
To get the Conway polynomial?, make one of the following substitutions:
To get the Alexander polynomial, make one of the following substitutions:
See the wikipedia page for the origin of the name.
Some fairly elementary discussion of the HOMFLY polynomial is given introductory texts such as
The original work was published as
- P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, and A. Ocneanu. (1985). A New Polynomial Invariant of Knots and Links Bulletin of the American Mathematical Society 12 (2): 239–246.
More recent work includes:
- A.Mironov, A.Morozov, An.Morozov, Character expansion for HOMFLY polynomials. I. Integrability and difference equations, arxiv/1112.5754
Revised on December 1, 2012 17:12:38
by Tim Porter